ABSTRACT
21.1. Spaces of germs of holomorphic mappings on compact subsets of Banach spaces were studied by Chae (1970, 1971) in connection with the Nachbin topology τω on the spaces Hθ(U;F)U of all holomorphic mappings on open subsets U of E for any holomorphic type θ. As a special case, when we take θ as the current type, we obtain Hθ(U;F)= H(U;F). For finite dimensional spaces, spaces of holomorphic germs were studied by Köther, Grothendieck, Dias, Martineau, and many others. The crux of this study is that through the interplay between the space H(U;F) and the spaces H(K;F) for all compact subsets K of U, we can obtain information about the Nachbin topology on H(U;F); for example, the completeness of H(U;F). The question of completeness of the topology τω on H(U;F) was investigated by Dineen, Chae, and Aron with some partial affirmative answers. It was a refinement of Chae’s idea to use spaces of holomorphic germs on compact subsets of U that eventually led Mujica (1979) to the general completeness theorem for (H(U;F),τω
